When the graph of a certain function $f(x)$ is shifted $2$ units to the right and stretched vertically by a factor of $2$ (meaning that all $y$-coordinates are doubled), the resulting figure is identical to the original graph. Given that $f(0)=0.1$, what is $f(10)$?

michaelcai
Aug 25, 2017

#1**+3 **

When the graph of a certain function f(x) is

shifted 2 units to the right

and stretched vertically by a factor of 2 (meaning that all y-coordinates are doubled), the resulting figure is identical to the original graph. Given that f(0)=0.1, what is f(10)?

f(x)=2f(x-2)

\(Given\qquad f(0)=0.1 \qquad find \;\;f(10)\\ f(x)=2f(x-2)\\ f(2)=2f(2-2)=2*f(0)=2*0.1=0.2\\ f(4)=2f(4-2)=2*f(2)=2*0.2=0.4\\ f(6)=2f(6-2)=2*f(4)=2*0.4=0.8\\ f(8)=2f(8-2)=2*f(6)=2*0.8=1.6\\ f(10)=2f(10-2)=2*f(8)=2*1.6=3.2\\~\\ f(10)=3.2 \)

Melody
Aug 26, 2017

#5**+1 **

I've been thinking more about this question.

When the graph of a certain function f(x) is

shifted 2 units to the right

and stretched vertically by a factor of 2 (meaning that all y-coordinates are doubled), the resulting figure is identical to the original graph. Given that f(0)=0.1, what is f(10)?

From my previous answer I have:

x | 0 | 2 | 4 | 6 | 8 | 10 |

f(x) | 0.1 | 0.2 | 0.4 | 0.8 | 1.6 | 3.2 |

These points satisfy the equation

\(f(x)=0.1*2^{(n/2)}\)

I graphed this to check it worked properly. It did :)

https://www.desmos.com/calculator/ugcl3yupfo

I am almost sure that if I think about it hard enough I could learn something more about transformations from this question ......

Melody
Aug 28, 2017